Montreal Geometric & Combinatorial Group Theory Seminar

Title:        “Measured laminations and extensions of homomorphisms

to free groups.
Date:        3:30PM, Monday, October 20, 2003
Place:       Room 1120, Burnside Hall, McGill University

Abstract:

Suppose H is a f.g. group that splits as H=A*_Z B over the

infinite cyclic group. Any homomorphism f:H->F to a free group induces a

natural family of homomorphisms f_t:H->F indexed by integers t and

obtained from f=f_0 by the t-fold Dehn twisting in the generator of Z.

Suppose H is embedded in a f.g. group G. Consider the set of indices t

for which f_t extends to G. What kinds of subsets of Z arise in this

way? We will show that the set is {\it eventually periodic}, i.e. for

some integer N>0 and for t_1,t_2 of sufficiently large absolute value,

if t_1 and t_2 are congruent mod N then either both f_{t_1}, f_{t_2}

extend or neither extends. There are generalizations when one considers

more complicated splittings and parameter sets.

This is joint work with Mark Feighn and is related to the recent work of

Zlil Sela about the Diopantine geometry over groups and Tarski's

problems.